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PROFESSOR: Hi.

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Welcome back to recitation.

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We've been talking about
applications of integration,

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including finding the areas
between curves.

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So I have here a nice little
region that I like.

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I think it's kind of cute.

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So it's the region between
y equals sine x and y

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equals cosine x.

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And, you know, those two curves
cross each other over

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and over again.

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They wrap around each other.

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So I'm just interested in the
region between the two

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consecutive points
where they cross.

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So here, you know, so at pi over
4 we have sine x equals--

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sine pi over 4 equals
cosine pi over 4.

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And at 5 pi over 4 they're also
equal again, down in the

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third quadrant there.

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So, the question is to compute
the area of this region that

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they bound between
those two points.

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So why don't you take a couple
of minutes, work through that,

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come back and we can work
through it together.

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All right, welcome back.

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So, from this picture it's
pretty easy to see what the

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region of integration is.

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I mean what the bounds
on x will be.

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As we said, x has to go the
left, I mean, you could--

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sorry, let me start over.

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You could do any two consecutive
intersection

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points you want, right?

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The area is the same in any case
because of the symmetry

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of these two functions.

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So, OK.

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But the first two that are
easiest for me to find are

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this pi over 4 and
this 5 pi over 4.

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So I'm going to do those two.

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If you wanted, you could have
done it with a different pair

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of consecutive points.

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But, once we've agreed that sort
of these first two are

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the ones I'm going to
do, then it's easy.

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I know that they're pi over
4 and 5 pi over 4.

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So that's the interval over
which I'm going to be

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integrating.

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And then, what I want to do is
just view this region as cut

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into a lot of little
rectangles.

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And I want to integrate the
height of those rectangles in

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order to get the area
of the whole region.

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So in this case, the upper curve
is y equals sine x and

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the lower curve is y
equals cosine of x.

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So the height of a little--

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you know if I drawn in a
little rectangle here--

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the height of that rectangle
is going to be sine x

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minus cosine x.

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Its width is dx.

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And then I add them all
up by integrating.

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So the area is just the integral
from pi over 4 to 5

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pi over 4 of sine x
minus cosine x dx.

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Top minus bottom to
get the height.

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If you did it backwards; if you
wrote cosine minus sine,

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what you would get is exactly
the negative of the area.

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So it's, all right, from pi
over 4 to 5, pi over 4.

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So now, we just have
to integrate this.

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So integral of sine, the
function whose derivative is

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sine is minus cosine x.

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And the function whose
derivative is cosine is sine.

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So it's minus sine x between
pi over 4, 5 pi over 4.

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And OK, so now, we just have
to plug in the values.

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So this is equal to minus cosine
5 pi over 4 minus sine

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5 pi over 4 minus minus
cosine pi over 4 minus

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sine pi over 4.

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And I'm sure you can work out
for yourself that this is

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equal to 2 times the square root
of 2 if I haven't botched

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anything terribly.

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So I'll end there.

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